We obtain some approximate identities whose accuracy depends on the bottom of the discrete spectrum of the Laplace-Beltrami operator in the automorphic setting and on the symmetries of the corresponding Maass wave forms. From the geometric point of view, the underlying Riemann surfaces are classical modular curves and Shimura curves.

We continue to investigate spt-type functions that arise from Bailey pairs. In this third paper on the subject, we proceed to introduce additional spt-type functions. We prove simple Ramanujan type congruences for these functions which can be explained by an spt-crank-type function. The spt-crank-type functions are actually defined first, with the spt-type functions coming from setting z = 1 in this definition. We find some of the spt-crank-type functions to have interesting representations as single...

Let $S$ be a fixed symmetric finite subset of $S{L}_d\left({𝒪}_K\right)$ that generates a Zariski dense subgroup of $S{L}_d\left({𝒪}_K\right)$ when we consider it as an algebraic group over $mathbbQ$ by restriction of scalars. We prove that the Cayley graphs of $S{L}_d({𝒪}_K/I)$ with respect to the projections of $S$ is an expander family if $I$ ranges over square-free ideals of ${𝒪}_K$ if $d=2$ and $K$ is an arbitrary numberfield, or if $d=3$ and $K=\mathbb{Q}$.

We note that every positive integer N has a representation as a sum of distinct members of the sequence ${d(n!)}_{n\ge 1}$, where d(m) is the number of divisors of m. When N is a member of a binary recurrence $u={uₙ}_{n\ge 1}$ satisfying some mild technical conditions, we show that the number of such summands tends to infinity with n at a rate of at least c₁logn/loglogn for some positive constant c₁. We also compute all the Fibonacci numbers of the form d(m!) and d(m₁!) + d(m₂)! for some positive integers m,m₁,m₂.

Let d ≥ 2 be an integer. In 2010, the second, third, and fourth authors gave necessary and sufficient conditions for the infinite products ${\prod }_{\genfrac{}{}{0pt}{}{k=1}{U_{d^k}\ne -a_i}}^{\infty }(1+\left(a_i\right)/\left(U_{d^k}\right))$ (i=1,...,m) or ${\prod }_{\genfrac{}{}{0pt}{}{k=1}{V_{d^k}\ne -a_i}}^{\infty }(1+\left(a_i\right)\left(V_{d^k}\right)$ (i=1,...,m) to be algebraically dependent, where $a_i$ are non-zero integers and $U_n$ and $V_n$ are generalized Fibonacci numbers and Lucas numbers, respectively. The purpose of this paper is to relax the condition on the non-zero integers $a_1,...,a_m$ to non-zero real algebraic numbers, which gives new cases where the infinite products above are algebraically dependent....